What is the Ratio of Internal to External Diameter for Equal Strength in Shafts?

[MMCS]

A mild steel solid shaft is to be replaced by a stainless steel hollow shaft of the same
outside diameter. Calculate the ratio of internal diameter to external diameter of the
hollow shaft for equal strength. Also find the percentage saving in weight if density of
mild steel is the same as stainless steel.Take maximum shear stress for Mild Steel as
34 MPa and that for Stainless Steel as 57 MPa

I assume that for the strength to be the same the max torsion should be the same so i use the max allowed shear stress in this equationrearranged for torsion and equate them both

(34MPa * (pi*R^4)/2)/R = (57MPa * (pi*(R-Ri)^4)/2)/R - Ri

RI = inside radius

Is this the correct step? how would i get a ratio from this?

 

[PhanthomJay]

A mild steel solid shaft is to be replaced by a stainless steel hollow shaft of the same
outside diameter. Calculate the ratio of internal diameter to external diameter of the
hollow shaft for equal strength. Also find the percentage saving in weight if density of
mild steel is the same as stainless steel.Take maximum shear stress for Mild Steel as
34 MPa and that for Stainless Steel as 57 MPa

I assume that for the strength to be the same the max torsion should be the same so i use the max allowed shear stress in this equationrearranged for torsion and equate them both

(34MPa * (pi*R^4)/2)/R = (57MPa * (pi*(R-Ri)^4)/2)/R - Ri

RI = inside radius

Is this the correct step? how would i get a ratio from this?

(R^4 - Ri^4) is not the same as (R - Ri)^4.
And in both cases, max shear stress occurs at the same distance fron the centroid of the circular cross section.

 

[MMCS]

Ok so

(34MPa * (pi*R^4)/2)/R = (57MPa * (pi/2*R^4)-(pi/2*Ri^4))/R

to simplify and multiply out pi terms i get

(-23*pi/2*r^4)/r = (57*pi/2*ri^4)/r

-36.128*r^4 = 89.535ri^4

so as a ratio i get

36.128/89.535 = 0.404

but i have the answer to be 0.8

 

[SteamKing]

You've got negative signs appearing and disappearing at random.

You should review basic algebra.

 

[MMCS]

You've got negative signs appearing and disappearing at random.

You should review basic algebra.

That is irrelevant. -36.128/89.535 produces the same ratio as 36.128/89.535.

 

[SteamKing]

And yet you have not solved this problem.

 

[MMCS]

And yet you have not solved this problem.

Another irrelevant response. I do not know how to solve the problem otherwise i wouldn't have posted the question, however, Your input had nothing to do with getting closer to the solution.

 

[PhanthomJay]

Ok so

(34MPa * (pi*R^4)/2)/R = (57MPa * (pi/2*R^4)-(pi/2*Ri^4))/R

to simplify and multiply out pi terms i get

(-23*pi/2*r^4)/r = (57*pi/2*ri^4)/r

-36.128*r^4 = 89.535ri^4

so as a ratio i get

36.128/89.535 = 0.404

the plus and minus signs heretofore not withstanding, you have calculated that (ri^4/r^4) = (ri/r)^4 = 0.404. So to calculate (ri/r), you need to take the 4th root of that number.

but i have the answer to be 0.8

which is correct.